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The Kamijou Faction is taken seriously? I don't even recall it ever being mentioned

Well, its hard to say how seriously its taken, but Lessar does mention it like its an actual thing. She could just be kidding, but the context makes it sound like she's serious.

Quote:

“Hmm? I’m not here because I was ordered to come by the British Royal Family, because I have a grudge against Fiamma of the Right, because I want to become a member of the Kamijou Faction, or anything else like that,” she responded in an exceedingly halfhearted way.

Chapter 2:
7 - Accelerator continues to know his shit, even as he enters a 'new world'
8 - Vento tries her best and Fiamma trolls everyone.
9 - Hamazura's God like luck takes a turn for the worse.
10 - Accelerator kicks ass, experiments like the scientist he is, and then gets his day ruined by a sexy troll
BiT 2 - Misaka stalks Kamijou via the internet.

I don't understand one thing. Accelerator's ability to automatic, why he needs to connect to the Misaka network to provide it?

In Volume 5 he turned off his automatic redirection (due to concentration) in order to delete the virus in LA. Desperate, Amai then shot Accelerator in the head,resulting in brain damage, hence he lost verbal and calculation ability.
This was solved by Heaven Canceller, by connecting him to the Misaka network to make up for his verbal and calculation ability.
Without the network, leading a normal life is impossible for him, as he can't walk, talk or understand human language properly (his brain can't keep up).

Quote:

This is a critical damage to Accelerator, as he has to first calculate the 'direction before redirection' and 'direction after redirection' before he can use his ability. Even if he could unconsciously use the 'redirection' ability, it's because he unknowingly calculated the simplest formula.
He won't be able to use his ability, even the most basic 'redirection'.

Because it's not automatic, it's a reflex- it still has to go through the Brain.

It only means he doesn't do it consciously.

For example, you don't need to think about blinking to do it.

most reflexes don't go through the brain its done at nerve points, but either way i think you're right. esper powers require calculation, it seems his calculating ability was so high that those calculations are automatic... Prolly under the involuntary action catagory, like blinking and such.

Chapter 3:
1 - Hamazura gets a tank dropped on his head, picks up his future Boss' slack and mans the fuck up.
2 - Sexy troll tries to mind break Accelerator. In the midst of breaking down, Accelerator continues to realize AC is fucking nuts.
3 - Cause it bears repeating; AC tech shits on everything, land, sea and sky.
4 - Hamazura can be one absolute badass.
5 - Kamijou has angst.
6 - Britain and France continue to battle. Carissa continues to be GAR.
7 - Worst continues to troll and Accelerator's mind breaks, but when Worst lies dying Accelerator still has the cognative functions to go "FUUUUUUUUUUUUUUCK THAAAAAAAAAAAT!!!" and be all batshit crazy hero.
8 - Misaka Worst realizes that mind breaking a man who could easily be voted into the Top 10 Scariest Mofo's of the series may not have been a good idea.
9 - More bad luck befalls Hamazura, but since he's still manned up from before, he starts trying to deal with it without being all wishy-washy.
10 - MindBroke Accelerator throws a tenper tantrum, cause his personal Hero didn't see him.
BtL 3 - Mikoto continues to stalk Kamijou, realizes that he's the Chairman's golden boy and feels the need to save him from the task force specifically designed to retrieve him. Well, that or its been 2 weeks and she can't bear not seeing him for any longer.

Hm, Laura better have a good plan, letting the power slide to the Science Side after WW3.

Well I'm sure she does, shes Laura after all and my Magical Queen.

__________________

He did not have extraordinary athletic ability.
He did not have the intellect or skill of a superhuman genius.
He did not have the fortune needed to run a nation.
He did not have the deductive ability to resolve any incident from his easy chair.
He did not have the beauty to ruin a country with a smile.
He did not have the political power to change history with a snap of his fingers.
He did not have the unbelievable power of a Level 5.
But…
Nevertheless… -

About Accelerator's brain, recovery probably is like it was with Touma in Vol. 1, where science alone can't recover the damaged brain. Index&co wanted to use magic on Touma to heal him, but IB would just negate it. Whether magic can help Accelerator is in the stars. He would have to allow an unfamiliar magician to perform magic on him as well as find a magician, but given Accelerator's nature from the darkness it is likely he'd never initiate contact with the magic side first.